Unified reconstruction of the Lyman-alpha power spectrum with Hamiltonian Monte Carlo

TL;DR

This work addresses reconstructing the Lyα forest’s three-dimensional power spectrum, $P_{3D}(k,\mu)$, from complementary two-point statistics ($P_{1D}$, $P_{\times}$, and $ξ_{\ell}$) by developing an analytic forward-model that leverages the integral relation between $P_{1D}$ and $P_{3D}$ and a physically motivated multipole framework. The authors implement a Hamiltonian Monte Carlo forward model with knot-based parametrization of the first few even multipoles and exploit ratio relations between multipoles to reduce degrees of freedom, enabling joint reconstruction across statistics. Tests on mock DESI-like data show that the $P_{3D}$ monopole can be reconstructed in 25 $k$-bins from $0.07$ to $1.8\,\mathrm{Mpc^{-1}}$ with ~13% average precision, and a joint analysis with $P_{\times}$ significantly improves monopole constraints. The approach provides a consistency-check pathway for $P_{3D}$ in real data and can inform future cosmological inferences, while explicitly noting the need to model observational systematics such as continuum distortions and metal absorbers.

Abstract

The complex geometry of the Ly$α$ forest data has motivated the use of various two-point statistics as alternatives to the three-dimensional power spectrum ($P_{\mathrm{3D}}$), which carries cosmological information in Fourier space. On large scales, the three-dimensional correlation function ($ξ_\mathrm{3D}$) has provided robust measurements of the baryon acoustic oscillation (BAO) scale at 150~Mpc. On smaller scales, the one-dimensional power spectrum, $P_{\mathrm{1D}}(k_\|)$, has been the primary tool for extracting information. At the same time, the cross-spectrum, $P_\times(θ, k_\|)$, has been introduced to incorporate angular information without the complications caused by survey window functions. We propose an analytical forward-modeling framework to reconstruct $P_{\mathrm{3D}}$ from all these observables, based on the mathematical relation between them and $P_{\mathrm{3D}}$. We demonstrate the performance of our method using a hypothetical mock data vector representative of future Dark Energy Spectroscopic Instrument (DESI) measurements. We show that the monopole of $P_{\mathrm{3D}}$ can be reconstructed in 25 $k$ bins between $0.07~\mathrm{Mpc}^{-1}$ and $1.8~\mathrm{Mpc}^{-1}$, achieving an average precision of $σ_P/P=13\%$ across the bins. Our method can serve as an intermediary for consistency checks, though it is not intended to replace direct $P_{\mathrm{3D}}$ estimation.

Figures (13)

• Figure 1: The ratio between higher order multipoles and the monopole based on the fitting function of Ly$\alpha$$P_{\mathrm{3D}}$. The quadrupole-to-monopole ratio is blue, whereas the hexadecapole-to-monopole ratio is orange. Solid lines follow our proposed fitting functions with the best-fit values, which can be found in the main text.
• Figure 2: The quadrupole-to-monopole ratio ( top) and the hexadecapole-to-monopole ratio ( bottom) for the redshift range $2<z<4$. Our fitting functions hold for the entire redshift range.
• Figure 3: Each multipole's contribution to $P_{\mathrm{1D}}$. The monopole and quadrupole compose the majority of $P_{\mathrm{1D}}$. We ignore the hexadecapole contribution for $P_{\mathrm{1D}}$ and $P_\times$ analyses.
• Figure 4: The reconstructed monopole ( top) and quadrupole ( bottom) from a noiseless $P_{\mathrm{1D}}$ data vector. While estimating the quadrupole independently ( blue triangles), the forward model cannot improve upon the priors. Leveraging the quadrupole-to-monopole ratio ( orange circles) can reconstruct the monopole in eight $k$ bins by yielding three times smaller error bars than the prior.
• Figure 5: ( Top) The reconstructed monopoles leveraging the quadrupole-to-monopole ratio from a $P_{\mathrm{1D}}$-only analysis ( blue triangles) and $P_{\mathrm{1D}}$$+P_\times$ joint analysis ( orange circles). The quadrupole is omitted because it contains no additional information beyond the monopole. ( Bottom) The error improvement factor between a $P_{\mathrm{1D}}$-only analysis and $P_{\mathrm{1D}}$$+P_\times$ joint analysis.